Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×(C22⋊C9)⋊C2 |
| commutator, cosocle, hypocenter | C3×Q8⋊C9 | C22 |
| Frattini | C2×C3 | D3×S4 |
| Fitting | C32×Q8 | C2×D3 |
| socle | C2×C32 | C2×S4 |
| prime radical | C3×#239 | D3 |
| LCS | G ▷ C3×Q8⋊C9 | |
| UCS | C1 ◁ C2 | |
135 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 135 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^2,3^3,2^3|a^2=h,b^2=h,c^3=e,f^2=h,g^2=h,b^a=bh,d^a=d^2,c^b=c^2e^2,e^b=e^2,f^b=gh,g^b=fh,f^c=g,g^c=fgh,g^f=gh42 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11722245, #13164184, #13164188, … |
| maximal normal subgroup of | C2×#11722245, #13164184, #13164188, … |
| maximal quotient of | #13146302, #13146303, #13146305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2212);